2 8 Se p 20 04 Technical notes on a 2 - d lattice O ( N ) model problem

Perturbation theory is the standard method to study quantum field theories in the small coupling regime. However, the interplay of perturbative expansion and the thermodynamic limit remains controversial. In particular, arguments were put forward that the infinite volume limit of perturbative coefficients does not give the correct infinite volume asymptotic perturbation expansion of asymptotically free theories [1]. In addition to the standard free (FBC), periodic (PBC), and Dirichlet (DBC) boundary conditions for the spin model considered, a novel boundary condition was introduced in [1], namely superinstanton boundary conditions (SIBC). The latter consist of Dirichlet conditions on the boundary of the system, and the additional freezing of one spin in the center of the sample. Perturbative coefficients were shown to have different thermodynamic limits for standard boundary conditions and SIBC. It was argued in [2] (see also [3]) that SIBC do not possess a well defined perturbation expansion, the third order coefficient being predicted to diverge in the infrared. Thus, perturbation theory was assumed to be consistent as one takes the V → ∞ limit for standard boundary conditions. This is a companion paper to [4] (so far the last contribution to the controversy, see citations therein), in which the volume dependence of perturbation theory coefficients for the O(N) vector model with different boundary conditions was investigated up to third order, confirming the points of [2] regarding independence of the infinite volume perturbative coefficients for ‘standard’ b.c. The aim of this paper is to describe in detail the method used to compute the perturbative coefficients in [4], and give a broader view of the results, including the IR divergence of SIBC correlators at third order.